Variants of Kato’s inequality are proved for general quasilinear elliptic operators L. As an outcome we show that, dealing with Liouville theorems for coercive equations of the type Lu = f (x, u,Du) on Ω ⊂ R^N , where f is such that f(x, t, ξ) t ≥ 0, the assumption that the possible solutions are nonnegative involves no loss of generality. Related consequences such as comparison principles and a priori bounds on solutions are also presented. An underlying structure throughout this work is the framework of Carnot groups.
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